Reason being, I'm looking into building the 3D Maneuver gear from Attack on Titan, though I think no ref is needed since I'm only inquiring on a best material for a basic, no frills rectangle, to ensure a sturdy base. I don't like how easy foamboard bends and dents, and then I heard it's possible to cut possible to cut balsa with an exacto knife, though most likely it'd have to be really thin. (Keep in mind I have only ever even touched balsa on two occassions...)My thinking is this negates any sturdiness it has over the foam board; and I'm half considering layering balsa over foam board.

(It's too early in the morning for me to make a nice paragraph of my situation so this'be in list form.)
It will be transported by plane at some point in a suitcase; I can't afford/don't have access to power tools or expensive materials; I live in Hawaii, so very limited selection of stuff; dimensions of it would hypothetically be around 23-26" long 12-9" wide (if this is important); and this will be spraypainted.

If any more experienced person could confirm or deny my sleep deprived thoughts I'd be so grateful, thank you in advance should I not say it later!

i probably have less experiance than you, but ill add in my 2 cents,
and dont be afraid to correct me if im wrong,

first, yes balsa is soft enought to be cut w/ a knife, and generally its easier w/ thiner pieces,
but your thinking of it like wood, while it is wood, its easier to think of it as the love child
of foamboard and a tree, id say balsa is closer to foam than it is other woods, such as pine

so you shouldnt worry about having to go thinner if you use balsa,
and it could be cut in the same way as foam (if you use power tools keep the piece weighted down)

my thinking is that foam and balsa are generally going to be about the same
except perhaps in terms of...

1, balsa wont "melt" if paint is put on it, but it might "drink" paint so you have to seal it (some foams will melt/dissolve depending on the paint)

2, balsa is wood(obvious) so if the grain is proper, it sould bend more than foam w/o breaking

3,oposite of 2, foam is more rigid, so even if it may crack easier when flexed,
it has its advantages in applications where you dont want flexing

Looking up ref photos, it looks like you're talking about these box-like objects at the characters' thighs.

By "foam board" I'm presuming you are referring to the sheet of foam that is sandwiched between 2 pieces of posterboard.

Generally, if you're planning to transport the thing through checked luggage, I'd advise you to go with something like plywood or MDF (Medium Density Fiberboard). However, the unbalanced nature of this prop would force you to very carefully build the harness mechanism that holds up the back. I'm pretty sure you have no interest in making anything so complicated. I'm a leathercrafter and a huge fan of overcomplicated props and I still wouldn't want to do something like that harness if I could avoid it.

I would urge you to either make this prop be your carryon item, or transporting it in an extremely durable hard-shell case of some sort (Possibly building a plywood crate, if that's even allowed in this day and age). Some cosplayers even go so far as buying their prop a ticket so they can strap it in.

Balsa wood splits very easily when flexed along the grain. The larger the dimensions of the board, the more these forces will be amplified when it is squished by other luggage pushing down on it. To resist this, you would either want to build the box around a solid styrofoam core, or fill the inside with rigid polyurethane expanding foam.

Another possibility would be to reinforce your piece with fiberglass+resin. Normally, I'll advise sticking to paper-mache, but that may be insufficient for such a large prop to survive transport. You should be able to get the materials for fiberglassing at a home improvement store or a marine supply store (I would suspect Hawaii has a lot of these). It's a nasty operation, it takes some practice, it must be done outdoors with a decent amount of safety equipment, it isn't cheap. But at least the results are very durable. If you go with fiberglassing, you could do it over whatever lightweight material you feel most comfortable with. Foam-core, pink foam, balsa. With pink foam or foam core, you merely need to seal the foam surfaces with acrylic paint first (Or use the more expensive epoxy resin).

If you do carryon your prop, I'd probably go with foam board. It's lighter than balsa, it cuts easier, it's cheaper, you don't have to worry about sealing off the grain to get a smooth paint job. If you can manage it, the box will be stronger if you build it with miter joints instead of butt joints. And again any portion of the box that is solid foam will be stronger than leaving it hollow, depending on how functional you want the prop to be. Cover the exposed foam edges with carefully cut pieces of posterboard or card stock. Fill any seam lines by pushing in wood putty in slight excess, letting it dry, and sanding it gently until smooth.

To protect it from random dents, which particularly effect the corners on such a piece, consider applying a couple layers of paper mache to the outside. It makes it stronger, but then you have to go through the laborious process of smoothing the surface, done with products like a slurry of paperclay, jointing compound or thinned wood-putty, repeatedly sanded smooth and flaws refilled until the surface looks flawless under bright light after painted with a coat of primer.